calc_theory: Find Theoretical Standardized Cumulants for Continuous...

In SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types

Description

This function calculates the theoretical mean, standard deviation, skewness, standardized kurtosis, and standardized fifth and sixth cumulants given either a Distribution name (plus up to 4 parameters) or a pdf (with specified lower and upper support bounds). The result can be used as input to find_constants or for data simulation.

Note: Due to the nature of the integration involved in calculating the standardized cumulants, the results are approximations. Greater accuracy can be achieved by increasing the number of subdivisions (sub) used in the integration process. However, the user may need to round the cumulants (i.e. using round(x, 8)) before using them in other functions (i.e. find_constants, calc_lower_skurt, nonnormvar1, rcorrvar, rcorrvar2) in order to achieve the desired results. For example, in order to ensure that skew is exactly 0 for symmetric distributions.

Usage

calc_theory(Dist = c("Benini", "Beta", "Beta-Normal", "Birnbaum-Saunders",
  "Chisq", "Dagum", "Exponential", "Exp-Geometric", "Exp-Logarithmic",
  "Exp-Poisson", "F", "Fisk", "Frechet", "Gamma", "Gaussian", "Gompertz",
  "Gumbel", "Kumaraswamy", "Laplace", "Lindley", "Logistic", "Loggamma",
  "Lognormal", "Lomax", "Makeham", "Maxwell", "Nakagami", "Paralogistic",
  "Pareto", "Perks", "Rayleigh", "Rice", "Singh-Maddala", "Skewnormal", "t",
  "Topp-Leone", "Triangular", "Uniform", "Weibull"), params = NULL,
  fx = NULL, lower = NULL, upper = NULL, sub = 1000)

Arguments

Dist	name of the distribution. The possible values are: "Benini", "Beta", "Beta-Normal", "Birnbaum-Saunders", "Chisq", "Exponential", "Exp-Geometric", "Exp-Logarithmic", "Exp-Poisson", "F", "Fisk", "Frechet", "Gamma", "Gaussian", "Gompertz", "Gumbel", "Kumaraswamy", "Laplace", "Lindley", "Logistic", "Loggamma", "Lognormal", "Lomax", "Makeham", "Maxwell", "Nakagami", "Paralogistic", "Pareto", "Perks", "Rayleigh", "Rice", "Singh-Maddala", "Skewnormal", "t", "Topp-Leone", "Triangular", "Uniform", "Weibull". Please refer to the documentation for each package (either stats-package, VGAM-package, or triangle) for information on appropriate parameter inputs.
params	a vector of parameters (up to 4) for the desired distribution (keep NULL if fx supplied instead)
fx	a pdf input as a function of x only, i.e. fx <- function(x) 0.5*(x-1)^2; must return a scalar (keep NULL if Dist supplied instead)
lower	the lower support bound for a supplied fx, else keep NULL
upper	the upper support bound for a supplied fx, else keep NULL
sub	the number of subdivisions to use in the integration; if no result, try increasing sub (requires longer computation time)

Value

A vector of the mean, standard deviation, skewness, standardized kurtosis, and standardized fifth and sixth cumulants

References

Headrick TC (2002). Fast Fifth-order Polynomial Transforms for Generating Univariate and Multivariate Non-normal Distributions. Computational Statistics & Data Analysis, 40(4):685-711. doi: 10.1016/S0167-9473(02)00072-5. (ScienceDirect)

Headrick TC, Kowalchuk RK (2007). The Power Method Transformation: Its Probability Density Function, Distribution Function, and Its Further Use for Fitting Data. Journal of Statistical Computation and Simulation, 77, 229-249. doi: 10.1080/10629360600605065.

Headrick TC, Sheng Y, & Hodis FA (2007). Numerical Computing and Graphics for the Power Method Transformation Using Mathematica. Journal of Statistical Software, 19(3), 1 - 17. doi: 10.18637/jss.v019.i03

Thomas W. Yee (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. https://CRAN.R-project.org/package=VGAM

Rob Carnell (2017). triangle: Provides the Standard Distribution Functions for the Triangle Distribution. R package version 0.11. https://CRAN.R-project.org/package=triangle

See Also

calc_fisherk, calc_moments, find_constants

Examples

options(scipen = 999)

# Pareto Distribution: params = c(alpha = scale, theta = shape)
calc_theory(Dist = "Pareto", params = c(1, 10))

# Generalized Rayleigh Distribution: params = c(alpha = scale, mu/sqrt(pi/2) = shape)
calc_theory(Dist = "Rayleigh", params = c(0.5, 1))

# Laplace Distribution: params = c(location, scale)
calc_theory(Dist = "Laplace", params = c(0, 1))

# Triangle Distribution: params = c(a, b)
calc_theory(Dist = "Triangular", params = c(0, 1))

Example output

Attaching package: 'SimMultiCorrData'

The following object is masked from 'package:stats':

    poly

       mean          sd        skew    kurtosis       fifth       sixth 
   1.111111    0.124226    2.811057   14.828571  130.208155 1808.899592 
      mean         sd       skew   kurtosis      fifth      sixth 
 0.4431135  0.2316257  0.6311107  0.2450893 -0.3131373 -0.8682880 
     mean        sd      skew  kurtosis     fifth     sixth 
 0.000000  1.414214  0.000000  3.000000  0.000000 30.000000 
                      mean                         sd 
 0.50000000000000000000000  0.20412414523193150861502 
                      skew                   kurtosis 
-0.00000000000000001197557 -0.59999670963988416261259 
                     fifth                      sixth 
 0.00000000000000013794433  1.71423587705926649960020 

SimMultiCorrData documentation built on May 2, 2019, 9:50 a.m.